Braverman–Kazhdan local $\rho$-Fourier transform conjecture

Let $G$ be a reductive group over a local field $F$, assume that there is an exact sequence

$$1\to G'\to G\to \mathbb{G}_m\to 1$$

with nontrivial homomorphism $\nu:G\to\mathbb{G}_m$, and let $\rho:{}^LG\to\operatorname{GL}(V_\rho)$ be irreducible with $\rho\circ\nu^\vee$ acting by scalar multiplication. Set

$$n_\rho=\langle 2\eta_G,\lambda_\rho\rangle+1,$$

where $2\eta_G$ is the sum of the positive roots and $\lambda_\rho$ is the highest weight of $\rho$, and define the local zeta integrals, basic function, invariant distribution, and $\rho$-Fourier transform as in the statement below.

Braverman–Kazhdan local $\rho$-Fourier transform conjecture. There exist a $\rho$-Schwartz space, a distinguished $\rho$-basic function, and an invariant distribution satisfying all the listed properties: meromorphic continuation and Langlands local $L$-factor generation for the zeta integrals; the stated basic-function identities; the Bernstein-centre and gamma-factor properties of the distribution; and an invertible $\rho$-Fourier transform

$$\mathcal{F}^{\rho}_{\psi}(\phi)(g)=|\nu(g)|^{-n_\rho}(J^{\rho}_{\psi}*\phi^\vee)(g)$$

that preserves the Schwartz space, satisfies

$$\mathcal{F}^{\rho}_{\psi}\circ\mathcal{F}^{\rho}_{\psi^{-1}}=\operatorname{Id},$$

is unitary on $L^2(G(F),|\nu(g)|^{n_\rho}\,\mathrm{d}g)$, fixes the basic function, and gives the functional equation

$$\mathcal{Z}(1-s,\mathcal{F}^{\rho}_{\psi}(\phi),\varphi_\pi^\vee)=\gamma(s,\pi,\rho,\psi)\mathcal{Z}(s,\phi,\varphi_\pi).$$

This is the local aspect of the Braverman–Kazhdan program. The statement packages analytic continuation, local factors, gamma factors, and Fourier duality into one proposed framework; the source presents it as a conjectural proposal rather than a theorem.